ora-0141
11.2 Pathwise loss and barycentric recovery
The qualifier “after expectation” is essential. It is the precise price of transporting full information through a one-coordinate observation channel.
If \(|A|\geq 2\), there is no map \(S:A\times [0,1]\to \mathcal L\) satisfying \(S(a,\ell (a))=\ell \) for every \(a\in A\) and every \(\ell \in \mathcal L\).
Fix an arm \(a\) and choose \(b\neq a\). Two loss vectors may agree at coordinate \(a\) and differ at coordinate \(b\). They generate the same observation \((a,\ell (a))\), so a deterministic map must return the same value on both, although exact reconstruction would require two different loss vectors.
The obstruction is informational, not algorithmic. No more ingenious estimator can make one realized scalar determine all unobserved coordinates. The distribution monad supplies a weaker but sufficient splitting: repaired evidence becomes exact after the observer \(\beta _{\mathcal L}\) averages over the sampling law.